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Dynamic Pricing in the Linear Valuation Model using Shape Constraints

Daniele Bracale, Moulinath Banerjee, Yuekai Sun, Kevin Stoll, Salam Turki

TL;DR

The paper tackles contextual dynamic pricing when the market-noise distribution $F_0$ is unknown. It introduces a tuning-parameter-free policy that leverages a shape constraint (monotonicity) on $F_0$ and Hölder smoothness, estimating $\theta_0$ by OLS and $F_0$ via antitonic (isotonic) regression using a doubling-epoch strategy. Theoretical results establish regret bounds that depend on the Hölder exponent $α$, with explicit rates for different regimes, and an efficient algorithmic implementation using the Pool Adjacent Violators Algorithm (PAVA). Empirically, the method demonstrates strong performance in simulations and a real-data application (Welltower Inc.), often outperforming Lipschitz-reliant baselines and tuning-parameter-based UCB approaches, while remaining parameter-free. The work also provides complexity analyses and discusses potential extensions such as optimal design and combining with existing methods.

Abstract

We propose a shape-constrained approach to dynamic pricing for censored data in the linear valuation model eliminating the need for tuning parameters commonly required by existing methods. Previous works have addressed the challenge of unknown market noise distribution $F_0$ using strategies ranging from kernel methods to reinforcement learning algorithms, such as bandit techniques and upper confidence bounds (UCB), under the assumption that $F_0$ satisfies Lipschitz (or stronger) conditions. In contrast, our method relies on isotonic regression under the weaker assumption that $F_0$ is $α$-Hölder continuous for some $α\in (0,1]$, for which we derive a regret upper bound. Simulations and experiments with real-world data obtained by Welltower Inc (a major healthcare Real Estate Investment Trust) consistently demonstrate that our method attains lower empirical regret in comparison to several existing methods in the literature while offering the advantage of being tuning-parameter free.

Dynamic Pricing in the Linear Valuation Model using Shape Constraints

TL;DR

The paper tackles contextual dynamic pricing when the market-noise distribution is unknown. It introduces a tuning-parameter-free policy that leverages a shape constraint (monotonicity) on and Hölder smoothness, estimating by OLS and via antitonic (isotonic) regression using a doubling-epoch strategy. Theoretical results establish regret bounds that depend on the Hölder exponent , with explicit rates for different regimes, and an efficient algorithmic implementation using the Pool Adjacent Violators Algorithm (PAVA). Empirically, the method demonstrates strong performance in simulations and a real-data application (Welltower Inc.), often outperforming Lipschitz-reliant baselines and tuning-parameter-based UCB approaches, while remaining parameter-free. The work also provides complexity analyses and discusses potential extensions such as optimal design and combining with existing methods.

Abstract

We propose a shape-constrained approach to dynamic pricing for censored data in the linear valuation model eliminating the need for tuning parameters commonly required by existing methods. Previous works have addressed the challenge of unknown market noise distribution using strategies ranging from kernel methods to reinforcement learning algorithms, such as bandit techniques and upper confidence bounds (UCB), under the assumption that satisfies Lipschitz (or stronger) conditions. In contrast, our method relies on isotonic regression under the weaker assumption that is -Hölder continuous for some , for which we derive a regret upper bound. Simulations and experiments with real-world data obtained by Welltower Inc (a major healthcare Real Estate Investment Trust) consistently demonstrate that our method attains lower empirical regret in comparison to several existing methods in the literature while offering the advantage of being tuning-parameter free.

Paper Structure

This paper contains 26 sections, 7 theorems, 63 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution.
  3. Related Works
  4. Notation
  5. Problem Setting
  6. Proposed Algorithm
  7. Estimation of $\theta_0$.
  8. Estimation of $S_0$.
  9. Exploitation.
  10. Complexity of the antitonic regression
  11. Regret Analysis
  12. Estimation of $\theta_0$
  13. Estimation of $S_0$ via antitonic regression
  14. Regret Upper Bound
  15. Simulations
  16. ...and 11 more sections

Key Result

Lemma 4.3

[fan2021policy] Let $n_k \triangleq |I_k|$ for simplicity of notation. Under Assumptions ass:1 and ass:bounded_iid_data, there exist $c_0,c_1>0$ depending only on absolute constants given in assumptions, such that for any episode $k$, as long as $n_{k} \geq c_0 d$, with probability at least $1- Q_{n where $C_{\theta_0}\triangleq \frac{8 \max \left\{R_{\mathcal{X}}, 1\right\}\left(R_{\mathcal{X}} R

Figures (7)

  • Figure 1: Picture of a general episode $\mathcal{J}_k$, $k=1,2,\dots, K$.
  • Figure 2: This plot shows the total expected regret (blue line) with $F_{0, \alpha}$, for $\alpha \in\{1/3,1/2,3/4\}$ in the first row, and a Gaussian, Laplace, and Cauchy c.d.f. in the second row (from the left to the right). We repeated the simulation $36$ times and displayed the corresponding $95 \%$ confidence intervals. The plot is in $\log _2$ - $\log _2$ scale to show the regret rate (empirical slope): a slope of $\eta$ indicated an $\mathcal{O}\left(T^\eta\right)$ regret. The black dashed line corresponds to our theoretical regret upper bound.
  • Figure 3: This plot in $\log_2-\log_2$ scale shows the cumulative regret over time up to $T= 4000$ for different values of $\tau_1 \in \{31,62,124,248\}$ and with $F_{0,1/3}$ for which theoretical regret rate is $0.86$. For each value of $\tau_1$, we repeated the simulation $36$ times and displayed the corresponding $95 \%$ confidence intervals. As we can see, the regret remains similar across different values of $\tau_1$ and the the empirical slopes are close to the theoretical regret rate.
  • Figure 4: Regret comparison in the simulation setting of tullii2024improved.
  • Figure 5: Residuals
  • ...and 2 more figures

Theorems & Definitions (16)

  • Lemma 4.3
  • Remark 4.4: The choice of the design points $w_t$
  • Remark 4.5: The difference between the conditional distributions of $y|(p,x)$ and $y|w$
  • Proposition 4.6
  • Theorem 4.8
  • Remark 4.9
  • Theorem 4.10
  • proof
  • Lemma 4.11
  • Remark 5.1: Dependence on $\tau_1$
  • ...and 6 more